MIMO different-factor partial-form model-free control with parameter self-tuning

ABSTRACT

The invention discloses a MIMO different-factor partial-form model-free control method with parameter self-tuning. In view of the limitations of the existing MIMO partial-form model-free control method with the same-factor structure, namely, at time k, different control inputs in the control input vector can only use the same values of penalty factor and step-size factors, the invention proposes a MIMO partial-form model-free control method with the different-factor structure, namely, at time k, different control inputs in the control input vector can use different values of penalty factors and/or step-size factors, which can solve control problems of strongly nonlinear MIMO systems with different characteristics between control channels widely existing in complex plants. Meanwhile, parameter self-tuning is proposed to effectively address the problem of time-consuming and cost-consuming when tuning the penalty factors and/or step-size factors. Compared with the existing method, the inventive method has higher control accuracy, stronger stability and wider applicability.

FIELD OF THE INVENTION

The present invention relates to the field of automatic control, andmore particularly to MIMO different-factor partial-form model-freecontrol with parameter self-tuning.

BACKGROUND OF THE INVENTION

In the fields of oil refining, petrochemical, chemical, pharmaceutical,food, paper, water treatment, thermal power, metallurgy, cement, rubber,machinery, and electrical industry, most of the controlled plants, suchas reactors, distillation columns, machines, devices, equipment,production lines, workshops and factories, are essentially MIMO systems(multi-input multi-output systems). Realizing the control of MIMOsystems with high accuracy, strong stability and wide applicability isof great significance to energy saving, consumption reduction, qualityimprovement and efficiency enhancement in industries. However, thecontrol problems of MIMO systems, especially of those with strongnonlinearities, have always been a major challenge in the field ofautomatic control.

MIMO partial-form model-free control method is one of the existingcontrol methods for MIMO systems. MIMO partial-form model-free controlmethod is a data-driven control method, which is used to analyze anddesign the controller depending only on the online measured input dataand output data instead of any mathematical model information of theMIMO controlled plant, and has good application prospects with conciseimplementation, low computational burden and strong robustness. Thetheoretical basis of MIMO partial-form model-free control method isproposed by Hou and Jin in Model Free Adaptive Control: Theory andApplications (Science Press, Beijing, China, 2013, p. 105), the controlscheme is given as follows:

${u(k)} = {{u\left( {k - 1} \right)} + \frac{{\Phi_{1}^{T}(k)}\left( {{\rho_{1}{e(k)}} - {\sum\limits_{p = 2}^{L}{\rho_{p}{\Phi_{p}(k)}\Delta{u\left( {k - p + 1} \right)}}}} \right)}{\lambda + {{\Phi_{1}(k)}}^{2}}}$

where u(k) is the control input vector at time k, u(k) [u₁(k), . . . ,u_(m)(k)]^(T), m is the total number of control inputs (m is a positiveinteger greater than 1), Δu(k)=u(k)−u(k−1); e(k) is the error vector attime k, e(k)=[e₁(k), . . . , e_(n)(k)]^(T), n is the total number ofsystem outputs (n is a positive integer); Φ(k) is the estimated value ofpseudo partitioned Jacobian matrix for MIMO system at time k, Φ_(p)(k)is the p-th block of Φ(k) (p is a positive integer, 1≤p≤L), ∥Φ₁(k)∥ isthe 2-norm of matrix Φ₁(k); λ is the penalty factor; ρ₁, . . . , ρ_(L)are the step-size factors; L is the control input length constant oflinearization and L is a positive integer.

The above-mentioned existing MIMO partial-form model-free control methodadopts the same-factor structure, namely, at time k, different controlinputs u₁(k), . . . , u_(m)(k) in the control input vector u(k) can onlyuse the same value of penalty factor λ, the same value of step-sizefactor ρ₁, . . . , and the same value of step-size factor ρ_(L).However, when applied to complex plants, such as strongly nonlinear MIMOsystems with different characteristics between control channels, theexisting MIMO partial-form model-free control method with thesame-factor structure is difficult to achieve ideal control performance,which restricts the popularization and application of MIMO partial-formmodel-free control method.

Therefore, in order to break the bottleneck of the existing MIMOpartial-form model-free control method with the same-factor structure,the present invention proposes a method of MIMO different-factorpartial-form model-free control with parameter self-tuning.

SUMMARY OF THE INVENTION

The present invention addresses the problems cited above, and provides amethod of MIMO different-factor partial-form model-free control withparameter self-tuning, the method comprising:

when a controlled plant is a MIMO system, namely a multi-inputmulti-output system, a mathematical formula for calculating the i-thcontrol input u_(i)(k) at time k using said method is as follows:

${u_{i}(k)} = {{u_{i}\left( {k - 1} \right)} + \frac{\sum\limits_{j = 1}^{n}{{\phi_{j,i,1}(k)}\left( {{\rho_{i,1}{e_{j}(k)}} - {\sum\limits_{p = 2}^{L}{\rho_{i,p}\left( {\sum\limits_{{iu} = 1}^{m}{{\phi_{j,{iu},p}(k)}\Delta{u_{iu}\left( {k - p + 1} \right)}}} \right)}}} \right)}}{\lambda_{i} + {{\Phi_{1}(k)}}^{2}}}$

where k is a positive integer; m is the total number of control inputsin said MIMO system, m is a positive integer greater than 1; n is thetotal number of system outputs in said MIMO system, n is a positiveinteger; i denotes the i-th of the total number of control inputs insaid MIMO system, i is a positive integer, 1≤i≤m; j denotes the j-th ofthe total number of system outputs in said MIMO system, j is a positiveinteger, 1≤j≤n; u_(i)(k) is the i-th control input at time k;Δu_(iu)(k)=u_(iu)(k)−u_(iu)(k−1), iu is a positive integer; e_(j)(k) isthe j-th error at time k, namely the j-th element in the error vectore(k)=[e₁(k), . . . , e_(n)(k)]^(T); Φ(k) is the estimated value ofpseudo partitioned Jacobian matrix for said MIMO system at time k,Φ_(p)(k) is the p-th block of Φ(k), ϕ_(j,i,p)(k) is the j-th row and thei-th column of matrix Φ_(p)(k), ∥Φ₁(k)∥ is the 2-norm of matrix Φ₁(k); pis a positive integer, 1≤p≤L; λ_(i) is the penalty factor for the i-thcontrol input; ρ_(i,p) is the p-th step-size factor for the i-th controlinput; L is the control input length constant of linearization and L isa positive integer;

for said MIMO system, traversing all values of i in the positive integerinterval [1, m], and calculating the control input vector u(k)=[u₁(k), .. . , u_(m)(k)]^(T) at time k using said method;

said method has a different-factor characteristic; said different-factorcharacteristic is that at least one of the following L+1 inequalitiesholds true for any two unequal positive integers i and x in the positiveinteger interval [1, m] during controlling said MIMO system by usingsaid method:

-   -   λ_(i)≠λ_(x); ρ_(i,1)≠ρ_(x,1); . . . ; ρ_(i,L)≠ρ_(x,L)

during controlling said MIMO system by using said method, performingparameter during controlling said MIMO system by using said method,performing parameter self-tuning on the parameters to be tuned in saidmathematical formula for calculating the control input vectoru(k)=[u₁(k), . . . , u_(m)(k)]^(T) at time k; said parameters to betuned comprise at least one of: penalty factors λ_(i), and step-sizefactors ρ_(i,1), . . . , ρ_(i,L) (i=1, . . . , m).

Said parameter self-tuning adopts neural network to calculate theparameters to be tuned in the mathematical formula of said control inputvector u(k)=[u₁(k), . . . , u_(m)(k)]^(T); when updating the hiddenlayer weight coefficients and output layer weight coefficients of saidneural network, the gradients at time k of said control input vectoru(k)=[u₁(k), . . . , u_(m)(k)]^(T) with respect to the parameters to betuned in their respective mathematical formula are used; the gradientsat time k of u_(i)(k) (i=1, . . . , m) in said control input vectoru(k)=[u₁(k), . . . , u_(m)(k)]^(T) with respect to the parameters to betuned in the mathematical formula of said u_(i)(k) comprise the partialderivatives at time k of u_(i)(k) with respect to the parameters to betuned in the mathematical formula of said u_(i)(k); the partialderivatives at time k of said u_(i)(k) with respect to the parameters tobe tuned in the mathematical formula of said u_(i)(k) are calculated asfollows:

when the parameters to be tuned in the mathematical formula of saidu_(i)(k) include penalty factor λ_(i), the partial derivative at time kof u_(i)(k) with respect to said penalty factor λ_(i) is:

$\frac{\partial{u_{i}(k)}}{\partial\lambda_{i}} = \frac{\sum\limits_{j = 1}^{n}{{\phi_{j,i,1}(k)}\left( {{\sum\limits_{p = 2}^{L}{\rho_{i,p}\left( {\sum\limits_{{iu} = 1}^{m}{{\phi_{j,{iu},p}(k)}\Delta{u_{iu}\left( {k - p + 1} \right)}}} \right)}} - {\rho_{i,1}{e_{j}(k)}}} \right)}}{\left( {\lambda_{i} + {{\Phi_{1}(k)}}^{2}} \right)^{2}}$

when the parameters to be tuned in the mathematical formula of saidu_(i)(k) include step-size factor ρ_(i,1), the partial derivative attime k of u_(i)(k) with respect to said step-size factor ρ_(i,1) is:

$\frac{\partial{u_{i}(k)}}{\partial\rho_{i,1}} = \frac{\sum\limits_{j = 1}^{n}{{\phi_{j,i,1}(k)}{e_{j}(k)}}}{\lambda_{i} + {{\Phi_{1}(k)}}^{2}}$

when the parameters to be tuned in the mathematical formula of saidu_(i)(k) include step-size factor ρ_(i,p) where 2≤p≤L, the partialderivative at time k of u_(i)(k) with respect to said step-size constantρ_(i,p) is:

$\frac{\partial{u_{i}(k)}}{\partial\rho_{i,p}} = {- \frac{\sum\limits_{j = 1}^{n}{{\phi_{j,i,1}(k)}\left( {\sum\limits_{{iu} = 1}^{m}{{\phi_{j,{iu},p}(k)}\Delta{u_{iu}\left( {k - p + 1} \right)}}} \right)}}{\lambda_{i} + {{\Phi_{1}(k)}}^{2}}}$

putting all partial derivatives at time k calculated by said u_(i)(k)with respect to the parameters to be tuned in the mathematical formulaof said u_(i)(k) into the set {gradient of u_(i)(k)}; for said MIMOsystem, traversing all values of i in the positive integer interval [1,m] and obtaining the set {gradient of u₁(k)}, . . . , set {gradient ofu_(m)(k)}, then putting them all into the set {gradient set}; said set{gradient set} is a set comprising all sets {{gradient of u₁(k)}, . . ., {gradient of u_(m)(k) }};

said parameter self-tuning adopts neural network to calculate theparameters to be tuned in the mathematical formula of the control inputvector u(k)=[u₁(k), . . . , u_(m)(k)]^(T); the inputs of said neuralnetwork comprise at least one of: elements in said set {gradient set},and elements in set {error set}; said set {error set} comprises at leastone of: the error vector e(k)=[e₁(k), . . . , e_(n)(k)]^(T), and errorfunction group of element e_(j)(k) (j=1, . . . , n) in said error vectore(k); said error function group of element e_(j)(k) comprises at leastone of: the accumulation of the j-th error at time k and all previoustimes

${\sum\limits_{t = 0}^{k}{e_{j}(t)}},$the first order backward difference of the j-th error e_(j)(k) at time ke_(j)(k)−e_(j)(k−1), the second order backward difference of the j-therror e_(j)(k) at time k e_(j)(k)−2e_(j)(k−1)+e_(j)(k−2), and high orderbackward difference of the j-th error e_(j)(k) at time k.

While adopting the above-mentioned technical scheme, the invention mayadopt or combine the following further technical schemes:

Said j-th error e_(j)(k) at time k is calculated by the j-th errorfunction; independent variables of said j-th error function comprise thej-th desired system output and the j-th actual system output.

Said j-th error function adopts at least one of: e_(j)(k)=y_(j)^(*)(k)−y_(j)(k), e_(j)(k)=y_(j) ^(*)(k+1)−y_(j)(k),e_(j)(k)=y_(j)(k)−y_(j) ^(*)(k), and e_(j)(k)=y_(j)(k)−y_(j) ^(*)(k+1),where y_(j) ^(*)(k) is the j-th desired system output at time k, y_(j)^(*)(k+1) is the j-th desired system output at time k+1, and y_(j)(k) isthe j-th actual system output at time k.

Said neural network is BP neural network; said BP neural network adoptsa single hidden layer structure, namely a three-layer network structure,comprising an input layer, a single hidden layer, and an output layer.

Aiming at minimizing a system error function, said neural network adoptsgradient descent method to update the hidden layer weight coefficientsand the output layer weight coefficients, where the gradients arecalculated by system error back propagation; independent variables ofsaid system error function comprise at least one of: elements in theerror vector e(k)=[e₁(k), . . . , e_(n)(k)]T, n desired system outputs,and n actual system outputs.

Said system error function is defined as

${{\sum\limits_{{jy} = 1}^{n}{a_{jy}{e_{jy}^{2}(k)}}} + {\sum\limits_{{iu} = 1}^{m}{b_{iu}\Delta{u_{iu}^{2}(k)}}}},$where e_(jy)(k) is the jy-th error at time k,Δu_(iu)(k)=u_(iu)(k)−u_(iu)(k−1), u_(iu)(k) is the iu-th control inputat time k, a_(jy) and b_(iu) are two constants greater than or equal tozero, jy and iu are two positive integers.

Said controlled plant comprises at least one of: a reactor, adistillation column, a machine, a device, a set of equipment, aproduction line, a workshop, and a factory.

The hardware platform for running said method comprises at least one of:an industrial control computer, a single chip microcomputer controller,a microprocessor controller, a field programmable gate array controller,a digital signal processing controller, an embedded system controller, aprogrammable logic controller, a distributed control system, a fieldbuscontrol system, an industrial control system based on internet ofthings, and an industrial internet control system.

The inventive MIMO different-factor partial-form model-free controlmethod with parameter self-tuning uses different penalty factors orstep-size factors for different control inputs in the control inputvector, which can solve control problems of strongly nonlinear MIMOsystems with different characteristics between control channels widelyexisting in complex plants. At the same time, parameter self-tuning isproposed to effectively address the problem of time-consuming andcost-consuming when tuning the penalty factors and/or step-size factors.Compared with the existing MIMO partial-form model-free control methodwith the same-factor structure, the inventive MIMO different-factorpartial-form model-free control method with parameter self-tuning hashigher control accuracy, stronger stability and wider applicability.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a schematic diagram according to the embodiments of theinvention;

FIG. 2 shows a structure diagram of the i-th BP neural network accordingto the embodiments of the invention;

FIG. 3 shows the tracking performance of the first system output whencontrolling the two-input two-output MIMO system in the first exemplaryembodiment by using the inventive MIMO different-factor partial-formmodel-free control method with parameter self-tuning;

FIG. 4 shows the tracking performance of the second system output whencontrolling the two-input two-output MIMO system in the first exemplaryembodiment by using the inventive MIMO different-factor partial-formmodel-free control method with parameter self-tuning;

FIG. 5 shows the control inputs when controlling the two-inputtwo-output MIMO system in the first exemplary embodiment by using theinventive MIMO different-factor partial-form model-free control methodwith parameter self-tuning;

FIG. 6 shows the changes of penalty factor when controlling thetwo-input two-output MIMO system in the first exemplary embodiment byusing the inventive MIMO different-factor partial-form model-freecontrol method with parameter self-tuning;

FIG. 7 shows the changes of the first step-size factor when controllingthe two-input two-output MIMO system in the first exemplary embodimentby using the inventive MIMO different-factor partial-form model-freecontrol method with parameter self-tuning;

FIG. 8 shows the changes of the second step-size factor when controllingthe two-input two-output MIMO system in the first exemplary embodimentby using the inventive MIMO different-factor partial-form model-freecontrol method with parameter self-tuning;

FIG. 9 shows the changes of the third step-size factor when controllingthe two-input two-output MIMO system in the first exemplary embodimentby using the inventive MIMO different-factor partial-form model-freecontrol method with parameter self-tuning;

FIG. 10 shows the tracking performance of the first system output whencontrolling the two-input two-output MIMO system in the first exemplaryembodiment by using the existing MIMO partial-form model-free controlmethod with the same-factor structure;

FIG. 11 shows the tracking performance of the second system output whencontrolling the two-input two-output MIMO system in the first exemplaryembodiment by using the existing MIMO partial-form model-free controlmethod with the same-factor structure;

FIG. 12 shows the control inputs when controlling the two-inputtwo-output MIMO system in the first exemplary embodiment by using theexisting MIMO partial-form model-free control method with thesame-factor structure;

FIG. 13 shows the tracking performance of the first system output whencontrolling the two-input two-output MIMO system of coal mill in thesecond exemplary embodiment;

FIG. 14 shows the tracking performance of the second system output whencontrolling the two-input two-output MIMO system of coal mill in thesecond exemplary embodiment;

FIG. 15 shows the first control input when controlling the two-inputtwo-output MIMO system of coal mill in the second exemplary embodiment;

FIG. 16 shows the second control input when controlling the two-inputtwo-output MIMO system of coal mill in the second exemplary embodiment;

FIG. 17 shows the changes of penalty factors λ₁ and λ₂ for two controlinputs when controlling the two-input two-output MIMO system of coalmill in the second exemplary embodiment;

FIG. 18 shows the changes of step-size factors ρ_(1,1), ρ_(1,2),ρ_(1,3), ρ_(1,4) for the first control input when controlling thetwo-input two-output MIMO system of coal mill in the second exemplaryembodiment;

FIG. 19 shows the changes of step-size factors ρ_(2,1), ρ_(2,2),ρ_(2,3), ρ_(2,4) for the second control input when controlling thetwo-input two-output MIMO system of coal mill in the second exemplaryembodiment.

DETAILED DESCRIPTION OF THE INVENTION

The invention is hereinafter described in detail with reference to theembodiments and accompanying drawings. It is to be understood that otherembodiments may be utilized and structural changes may be made withoutdeparting from the scope of the invention.

FIG. 1 shows a schematic diagram according to the embodiments of theinvention. For a MIMO system with m inputs (m is a positive integergreater than 1) and n outputs (n is a positive integer), the MIMOdifferent-factor partial-form model-free control method is adopted tocontrol the system; set the control input length constant oflinearization L of the MIMO different-factor partial-form model-freecontrol method where L is a positive integer. For the i-th control inputu_(i)(k) (i=1, . . . , m), the parameters in the mathematical formulafor calculating u_(i)(k) using the MIMO different-factor partial-formmodel-free adaptive control method include penalty factor λ_(i) andstep-size factors ρ_(i,1), . . . , ρ_(i,L); choose the parameters to betuned in the mathematical formula of u_(i)(k), which are part or all ofthe parameters in the mathematical formula of u_(i)(k), including atleast one of the penalty factor λ_(i), and step-size factors ρ_(i,1), .. . , ρ_(i,L); in the schematic diagram of FIG. 1 , the parameters to betuned in the mathematical formula of all control inputs u_(i)(k) (i=1, .. . , m) are the penalty factors λ_(i) and step-size factors ρ_(i,1), .. . , ρ_(i,L); the parameters to be tuned in the mathematical formula ofu_(i)(k) are calculated by the i-th BP neural network.

FIG. 2 shows a structure diagram of the i-th BP neural network accordingto the embodiments of the invention; BP neural network can adopt asingle hidden layer structure or a multiple hidden layers structure; forthe sake of simplicity, BP neural network in the diagram of FIG. 2adopts the single hidden layer structure, namely a three-layer networkstructure comprising an input layer, a single hidden layer and an outputlayer; set the number of input layer nodes, hidden layer nodes andoutput layer nodes of the i-th BP neural network; the number of inputlayer nodes of the i-th BP neural network is set to m×(Ly+Lu+1)+n×3,m×(Ly+Lu+1) of which are the elements

$\left\{ {\frac{\partial{u_{iu}\left( {k - 1} \right)}}{\partial\lambda_{iu}},\ \frac{\partial{u_{iu}\left( {k - 1} \right)}}{\partial\rho_{{iu},1}}\ ,\ldots,\ \frac{\partial{u_{iu}\left( {k - 1} \right)}}{\partial\rho_{{iu},L}}} \right\}$(iu=1, . . . , m) in set {gradient set}, and the other n×3 are theelements

$\left\{ {{e_{jy}(k)},{\sum\limits_{t = 0}^{k}{e_{jy}(t)}},} \right.$e_(jy)(k)−e_(jy)(k−1)} (jy=1, . . . , n) in set {error set}; the numberof output layer nodes of the i-th BP neural network is no less than thenumber of parameters to be tuned in the mathematical formula ofu_(i)(k); in FIG. 2 , the number of parameters to be tuned in themathematical formula of u_(i)(k) is L+1, which are the penalty factorλ_(i) and step-size factors ρ_(i,1), . . . , ρ_(i,L); detailed updatingprocess of hidden layer weight coefficients and output layer weightcoefficients of the i-th BP neural network is as follows: in FIG. 2 ,aiming at minimizing the system error function

$\sum\limits_{{jy} = 1}^{n}{e_{y}^{2}(k)}$with all contributions of n errors comprehensively considered, thegradient descent method is used to update the hidden layer weightcoefficients and the output layer weight coefficients of the i-th BPneural network, where the gradients is calculated by system error backpropagation; in the process of updating the hidden layer weightcoefficients and the output layer weight coefficients of the i-th BPneural network, the elements in set {gradient set}, comprising the set{gradient of u₁(k)}, . . . , set {gradient of u_(m)(k)}, are used,namely the gradients at time

$k\left\{ {\frac{\partial{u_{iu}(k)}}{\partial\lambda_{iu}},\ \frac{\partial{u_{iu}(k)}}{\partial{\rho_{iu}}_{1}},\ldots,\ \frac{\partial{u_{iu}(k)}}{\partial\rho_{{iu},L}}} \right\}\left( {{{iu} = 1},\ldots,m} \right)$of the control input vector u(k)=[u₁(k), . . . , u_(m)(k)]^(T) withrespect to parameters to be tuned in their respective mathematicalformula.

In combination with the above description of FIG. 1 and FIG. 2 , theimplementation steps of the technical scheme in the present inventionare further explained as follows:

mark the current moment as time k; define the difference between thej-th desired system output y_(j) ^(*)(k) and the j-th actual systemoutput y_(j)(k) as the j-th error e_(j)(k); traverse all values of j inthe positive integer interval [1, n] and obtain the error vectore(k)=[e₁(k), . . . , e_(n)(k)]^(T), then put them all into the set{error set}; take the elements

$\left\{ {\frac{\partial{u_{iu}(k)}}{\partial\lambda_{iu}},\ \frac{\partial{u_{iu}(k)}}{\partial{\rho_{iu}}_{1}},\ \text{...},\ \frac{\partial{u_{iu}(k)}}{\partial\rho_{{iu},L}}} \right\}\left( {{{iu} = 1},\ldots,m} \right)$in set {gradient set} and the elements

$\left\{ {{e_{jy}(k)},{\sum\limits_{t = 0}^{k}{e_{jy}(t)}},{{e_{jy}(k)} - {e_{jy}\left( {k - 1} \right)}}} \right\}\left( {{{jy} = 1},\ldots,n} \right)$in set {error set} as the inputs of the i-th BP neural network; obtainthe parameters to be tuned in the mathematical formula for calculatingu_(i)(k) using the MIMO different-factor partial-form model-free controlmethod by the output layer of the i-th BP neural network through forwardpropagation; based on the error vector e(k) and the parameters to betuned in the mathematical formula of u_(i)(k), calculate the i-thcontrol input u_(i)(k) at time k using the MIMO different-factorpartial-form model-free adaptive control method; traverse all values ofi in the positive integer interval [1, m] and obtain the control inputvector u(k)=[u₁(k), . . . , u_(m)(k)]^(T) at time k; for u_(i)(k) in thecontrol input vector u(k), calculate all partial derivatives of u_(i)(k)with respect to the parameters to be tuned in the mathematical formula,and put them all into the set {gradient of u_(i)(k)}; traverse allvalues of i in the positive integer interval [1, m] and obtain the set{gradient of u₁(k)}, . . . , set {gradient of u_(m)(k)}, and put themall into the set {gradient set}; then, aiming at minimizing the systemerror function

$\sum\limits_{{jy} = 1}^{n}{e_{jy}^{2}(k)}$with all contributions of n errors comprehensively considered and usingthe gradients

$\left\{ {\frac{\partial{u_{iu}(k)}}{\partial\lambda_{iu}},\frac{\partial{u_{iu}(k)}}{\partial\rho_{{iu},1}},\ldots,\frac{\partial{u_{iu}(k)}}{\partial\rho_{{iu},L}}} \right\}$(iu=1, . . . , m) in set {gradient set}, update the hidden layer weightcoefficients and the output layer weight coefficients of the i-th BPneural network using the gradient descent method, where the gradients iscalculated by system error back propagation; traverse all values of i inthe positive integer interval [1, m] and update the hidden layer weightcoefficients and the output layer weight coefficients of all m BP neuralnetworks; obtain the n actual system outputs at next time by applyingthe control input vector u(k) into the controlled plant, and then repeatthe steps described in this paragraph for controlling the MIMO system atnext sampling time.

Two exemplary embodiments of the invention are given for furtherexplanation.

The First Exemplary Embodiment

A two-input two-output MIMO system, which has the complexcharacteristics of non-minimum phase nonlinear system, is adopted as thecontrolled plant, and it belongs to the MIMO system that is particularlydifficult to control:

${{y_{1}(k)} = {\frac{{2.5}{y_{1}\left( {k - 1} \right)}{y_{1}\left( {k - 2} \right)}}{1 + {y_{1}^{2}\left( {k - 1} \right)} + {y_{2}^{2}\left( {k - 2} \right)} + {y_{1}^{2}\left( {k - 3} \right)}} + {0.09{u_{1}\left( {k - 1} \right)}{u_{1}\left( {k - 2} \right)}} + {1.2{u_{1}\left( {k - 1} \right)}} + {1.6{u_{1}\left( {k - 3} \right)}} + {{0.5}{u_{2}\left( {k - 1} \right)}} + {0.7{\sin\left( {{0.5}\left( {{y_{1}\left( {k - 1} \right)} + {y_{1}\left( {k - 2} \right)}} \right)} \right)}{\cos\left( {{0.5}\left( {{y_{1}\left( {k - 1} \right)} + {y_{1}\left( {k - 2} \right)}} \right)} \right)}}}}{{y_{2}(k)} = {\frac{5{y_{2}\left( {k - 1} \right)}{y_{2}\left( {k - 2} \right)}}{1 + {y_{2}^{2}\left( {k - 1} \right)} + {y_{1}^{2}\left( {k - 2} \right)} + {y_{2}^{2}\left( {k - 3} \right)}} + {u_{2}\left( {k - 1} \right)} + {{1.1}{u_{2}\left( {k - 2} \right)}} + {1.4{u_{2}\left( {k - 3} \right)}} + {{0.5}{u_{1}\left( {k - 1} \right)}}}}$

The desired system outputs y*(k) are as follows:y ₁ ^(*)(k)=5 sin(k/50)+2 cos(k/20)y ₂ ^(*)(k)=2 sin(k/50)+5 cos(k/20)

In this embodiment, m=n=2.

The control input length constant of linearization L is usually setaccording to the complexity of the controlled plant and the actualcontrol performance, generally between 1 and 10, while small Lu willaffect the control performance and large L will lead to massivecalculation, so it is usually set to 3 or 5; in this embodiment, L=3.

In view of the above exemplary embodiment, five experiments are carriedout for comparison and verification. In order to compare the controlperformance of the five experiments clearly, root mean square error(RMSE) is adopted as the control performance index for evaluation:

${{RMSE}\left( e_{j} \right)} = \sqrt{\frac{1}{N}{\sum\limits_{k = 1}^{N}{e_{j}^{2}(k)}}}$

where e_(j)(k)=y_(j) ^(*)(k)−y_(j)(k), y_(j) ^(*)(k) is the j-th desiredsystem output at time k, y_(j)(k) is the j-th actual system output attime k. The smaller the value of RMSE(e_(j)) is, the smaller the errorbetween the j-th actual system output and the j-th desired system outputis, and the better the control performance gets.

The hardware platform for running the inventive control method is theindustrial control computer.

The first experiment (RUN1): the number of input layer nodes of thefirst BP neural network and the second BP neural network is both set to14, 8 of which are the elements

$\left\{ {\frac{\partial{u_{1}\left( {k - 1} \right)}}{\partial\lambda_{1}},\frac{\partial{u_{1}\left( {k - 1} \right)}}{\partial\rho_{1,1}},\frac{\partial{u_{1}\left( {k - 1} \right)}}{\partial\rho_{1,2}},\frac{\partial{u_{1}\left( {k - 1} \right)}}{\partial\rho_{1,3}},\frac{\partial{u_{2}\left( {k - 1} \right)}}{\partial\lambda_{2}},\frac{\partial{u_{2}\left( {k - 1} \right)}}{\partial\rho_{2,1}},\frac{\partial{u_{2}\left( {k - 1} \right)}}{\partial\rho_{2,2}},\frac{\partial{u_{2}\left( {k - 1} \right)}}{\partial\rho_{2,3}}} \right\}$in set {gradient set}, and the other 6 are the elements

$\left\{ {{e_{1}(k)},{\sum\limits_{t = 0}^{k}{e_{1}(t)}},{{e_{1}(k)} - {e_{1}\left( {k - 1} \right)}},{e_{2}(k)},{\sum\limits_{t = 0}^{k}{e_{2}(t)}},{{e_{2}(k)} - {e_{2}\left( {k - 1} \right)}}} \right\}$in set {error set}; the number of hidden layer nodes of the first BPneural network and the second BP neural network is both set to 6; thenumber of output layer nodes of the first BP neural network and thesecond BP neural network is both set to 4, where the first BP neuralnetwork outputs penalty factor λ₁ and step-size factors ρ_(1,1),ρ_(1,2), ρ_(1,3), and the second BP neural network outputs penaltyfactor λ₂ and step-size factors ρ_(2,1), ρ_(2,2), ρ_(2,3); the inventiveMIMO different-factor partial-form model-free control method withparameter self-tuning is adopted to control the above two-inputtwo-output MIMO system; the tracking performance of the first systemoutput and second system output are shown in FIG. 3 and FIG. 4 ,respectively, and the control inputs are shown in FIG. 5 ; FIG. 6 showsthe changes of penalty factor, FIG. 7 shows the changes of the firststep-size factor, FIG. 8 shows the changes of the second step-sizefactor, and FIG. 9 shows the changes of the third step-size factor;evaluate the control method from the control performance indexes: theRMSE(e₁) of the first system output in FIG. 3 is 0.3781, and theRMSE(e₂) of the second system output in FIG. 4 is 0.7201; evaluate thecontrol method from the different-factor characteristic: the changes ofpenalty factor in FIG. 6 basically do not overlap, indicating that thedifferent-factor characteristic for penalty factor is significant whencontrolling the above two-input two-output MIMO system, and the changesof step-size factors in FIG. 7, 8, 9 basically do not overlap,indicating that the different-factor characteristic for step-sizefactors are significant when controlling the above two-input two-outputMIMO system.

The second experiment (RUN2): the number of input layer nodes of thefirst BP neural network and the second BP neural network is both set to8, all of which are the elements

$\left\{ {\frac{\partial{u_{1}\left( {k - 1} \right)}}{\partial\lambda_{1}},\frac{\partial{u_{1}\left( {k - 1} \right)}}{\partial\rho_{1,1}},\frac{\partial{u_{1}\left( {k - 1} \right)}}{\partial\rho_{1,2}},\frac{\partial{u_{1}\left( {k - 1} \right)}}{\partial\rho_{1,3}},\frac{\partial{u_{2}\left( {k - 1} \right)}}{\partial\lambda_{2}},\frac{\partial{u_{2}\left( {k - 1} \right)}}{\partial\rho_{2,1}},\frac{\partial{u_{2}\left( {k - 1} \right)}}{\partial\rho_{2,2}},\frac{\partial{u_{2}\left( {k - 1} \right)}}{\partial\rho_{2,3}},} \right\}$in set {gradient set}; the number of hidden layer nodes of the first BPneural network and the second BP neural network is both set to 6; thenumber of output layer nodes of the first BP neural network and thesecond BP neural network is both set to 4, where the first BP neuralnetwork outputs penalty factor λ₁ and step-size factors ρ_(1,1),ρ_(1,2), ρ_(1,3), and the second BP neural network outputs penaltyfactor λ₂ and step-size factors ρ_(2,1), ρ_(2,2), ρ_(2,3); the inventiveMIMO different-factor partial-form model-free control method withparameter self-tuning is adopted to control the above two-inputtwo-output MIMO system; evaluate the control method from the controlperformance indexes: the RMSE(e₁) of the first system output is 0.4761,and the RMSE(e₂) of the second system output is 0.8876.

The third experiment (RUN3): the number of input layer nodes of thefirst BP neural network and the second BP neural network is both set to6, all of which are the elements

$\left\{ {{e_{1}(k)},{\sum\limits_{t = 0}^{k}{e_{1}(t)}},{{e_{1}(k)} - {e_{1}\left( {k - 1} \right)}},{e_{2}(k)},{\sum\limits_{t = 0}^{k}{e_{2}(t)}},{{e_{2}(k)} - {e_{2}\left( {k - 1} \right)}}} \right\}$in set {error set}; the number of hidden layer nodes of the first BPneural network and the second BP neural network is both set to 6; thenumber of output layer nodes of the first BP neural network and thesecond BP neural network is set to 4, where the first BP neural networkoutputs penalty factor λ₁ and step-size factors, ρ_(1,1), ρ_(1,2),ρ_(1,3), and the second BP neural network outputs penalty factor λ₂ andstep-size factors ρ_(2,1), ρ_(2,2), ρ_(2,3); the inventive MIMOdifferent-factor partial-form model-free control method with parameterself-tuning is adopted to control the above two-input two-output MIMOsystem; evaluate the control method from the control performanceindexes: the RMSE(e₁) of the first system output is 0.5256, and theRMSE(e₂) of the second system output is 1.1019.

The fourth experiment (RUN4): the penalty factors λ₁, λ₂ and step-sizefactors ρ_(1,1), ρ_(1,2), ρ_(1,3) are fixed, and only the step-sizefactors ρ_(2,1), ρ_(2,2), ρ_(2,3) for the second control input arechosen for the parameters to be tuned, therefore, only one BP neuralnetwork is adopted here; the number of input layer nodes of the BPneural network is set to 6, all of which are the elements

$\left\{ {{e_{1}(k)},{\sum\limits_{t = 0}^{k}{e_{1}(t)}},{{e_{1}(k)} - {e_{1}\left( {k - 1} \right)}},{e_{2}(k)},{\sum\limits_{t = 0}^{k}{e_{2}(t)}},{{e_{2}(k)} - {e_{2}\left( {k - 1} \right)}}} \right\}$in set {error set}; the number of hidden layer nodes of the BP neuralnetwork is set to 6; the number of output layer nodes of the BP neuralnetwork is set to 3, where the outputs are step-size factors ρ_(2,1),ρ_(2,2), ρ_(2,3); the inventive MIMO different-factor partial-formmodel-free control method with parameter self-tuning is adopted tocontrol the above two-input two-output MIMO system; evaluate the controlmethod from the control performance indexes: the RMSE(e₁) of the firstsystem output is 0.5876, and the RMSE(e₂) of the second system output is1.1861.

The fifth experiment (RUN5): the existing MIMO partial-form model-freecontrol method is adopted control the above two-input two-output MIMOsystem; set the penalty factor λ=0.01, and the step-size factorsρ₁=ρ₂=ρ₃=0.50; the tracking performance of the first system output andthe second system output are shown in FIG. 10 and FIG. 11 ,respectively, and the control inputs are shown in FIG. 12 ; evaluate thecontrol method from the control performance indexes: the RMSE(e₁) of thefirst system output is 0.7235, and the RMSE(e₂) of the second systemoutput is 2.1182.

The comparison results of control performance indexes of the fiveexperiments are shown in Table 1; the results of the first experiment tothe fourth experiment (RUN1, RUN2, RUN3, RUN4) using the inventivecontrol method are superior to those of the fifth experiment (RUN5)using the existing MIMO partial-form model-free control method with thesame-factor structure, and have a significant improvement, indicatingthat the inventive MIMO different-factor partial-form model-free controlmethod with parameter self-tuning has higher control accuracy, strongerstability and wider applicability.

TABLE 1 Comparison Results of The Control Performance The first systemoutput The second system output RMSE(e₁) Improvement RMSE(e₂)Improvement RUN1 0.3781 47.740% 0.7201 66.004% RUN2 0.4761 34.195%0.8876 58.096% RUN3 0.5256 27.353% 1.1019 47.979% RUN4 0.5876 18.784%1.1861 44.004% RUN5 0.7235 Baseline 2.1182 Baseline

The Second Exemplary Embodiment

A coal mill is a very important set of equipment that pulverizes rawcoal into fine powder, providing fine powder for the pulverized coalfurnace. Realizing the control of coal mill with high accuracy, strongstability and wide applicability is of great significance to ensure thesafe and stable operation of thermal power plant.

The two-input two-output MIMO system of coal mill, which has the complexcharacteristics of nonlinearity, strong coupling and time-varying, isadopted as the controlled plant, and it belongs to the MIMO system thatis particularly difficult to control. Two control inputs u₁(k) and u₂(k)of the coal mill are hot air flow (controlled by the opening of hot airgate) and recycling air flow (controlled by the opening of recycling airgate), respectively. Two system outputs y₁(k) and y₂(k) of the coal millare outlet temperature (° C.) and inlet negative pressure (Pa),respectively. The initial conditions of the coal mill are: u₁(0)=80%,u₂(0)=40%, y₁(0)=70° C., y₂(0)=−400 Pa. At the 50th second, in order tomeet the needs of on-site conditions adjustment in thermal power plant,the desired system output y₁*(50) is adjusted from 70° C. to 80° C., andthe desired system output y₂*(k) is required to remain unchanged at −400Pa. In view of the above typical conditions in the thermal power plant,two experiments are carried out for comparison and verification. In thisembodiment, m=n=2, the control input length constant of linearization Lis set to 4. The hardware platform for running the inventive controlmethod is the industrial control computer.

The sixth experiment (RUN6): the number of input layer nodes of thefirst BP neural network and the second BP neural network is both set to16, 10 of which are the elements

$\left\{ {\frac{\partial{u_{1}\left( {k - 1} \right)}}{\partial\lambda_{1}},\frac{\partial{u_{1}\left( {k - 1} \right)}}{\partial\rho_{1,1}},\frac{\partial{u_{1}\left( {k - 1} \right)}}{\partial\rho_{1,2}},\frac{\partial{u_{1}\left( {k - 1} \right)}}{\partial\rho_{1,3}},\frac{\partial{u_{1}\left( {k - 1} \right)}}{\partial\rho_{1,4}},\frac{\partial{u_{2}\left( {k - 1} \right)}}{\partial\lambda_{2}},\frac{\partial{u_{2}\left( {k - 1} \right)}}{\partial\rho_{2,1}},\frac{\partial{u_{2}\left( {k - 1} \right)}}{\partial\rho_{2,2}},{\frac{\partial{u_{2}\left( {k - 1} \right)}}{\partial\rho_{2,3}}\frac{\partial{u_{2}\left( {k - 1} \right)}}{\partial\rho_{2,4}}},} \right\}$in set {gradient set}, and the other 6 are the elements

$\left\{ {{e_{1}(k)},{\sum\limits_{t = 0}^{k}{e_{1}(t)}},{{e_{1}(k)} - {e_{1}\left( {k - 1} \right)}},{e_{2}(k)},{\sum\limits_{t = 0}^{k}{e_{2}(t)}},{{e_{2}(k)} - {e_{2}\left( {k - 1} \right)}}} \right\}$in set {error set}; the number of hidden layer nodes of the first BPneural network and the second BP neural network is both set to 20; thenumber of output layer nodes of the first BP neural network and thesecond BP neural network is both set to 5, where the first BP neuralnetwork outputs penalty factor λ₁ and step-size factors ρ_(1,1),ρ_(1,2), ρ_(1,3), ρ_(1,4), and the second BP neural network outputspenalty factor λ₂ and step-size factors ρ_(1,2), ρ_(1,2), ρ_(2,3),ρ_(2,4); the inventive MIMO different-factor partial-form model-freecontrol method with parameter self-tuning is adopted to control theabove two-input two-output MIMO system; the tracking performance of thefirst output is shown as RUN6 in FIG. 13 , the tracking performance ofthe second output is shown as RUN6 in FIG. 14 , the first control inputis shown as RUN6 in FIG. 15 , and the second control input is shown asRUN6 in FIG. 16 ; FIG. 17 shows the changes of penalty factors λ₁ and λ₂for two control inputs, FIG. 18 shows the changes of step-size factorsρ_(1,1), ρ_(1,2), ρ_(1,3), ρ_(1,4) for the first control input, and FIG.19 shows the changes of step-size factors ρ_(2,1), β_(2,2), ρ_(2,3),ρ_(2,4) for the second control input; evaluate the control method fromthe control performance indexes: the RMSE(e₁) of the first systemoutput, RUN6 in FIG. 13 , is 2.0549, and the RMSE(e₂) of the secondsystem output, RUN6 in FIG. 14 , is 0.1554; evaluate the control methodfrom the different-factor characteristic: the changes of penalty factorsfor two control inputs in FIG. 17 do not overlap at all, indicating thatthe different-factor characteristic for penalty factor is significantwhen controlling the above two-input two-output MIMO system, and thechanges of step-size factors for two control inputs in FIG. 18 and FIG.19 basically do not overlap, indicating that the different-factorcharacteristic for step-size factors are significant when controllingthe above two-input two-output MIMO system.

The seventh experiment (RUN7): the MIMO different-factor partial-formmodel-free control method with fixed parameters is adopted to controlthe above two-input two-output MIMO system; set the parameters value forcalculating the first control input: the penalty factor λ₁=0.05, thestep-size factors ρ_(1,1)=1.98, ρ_(1,2)=1.32, ρ_(1,3)=1.14,ρ_(1,4)=1.11; set the parameters value for calculating the secondcontrol input: the penalty factor λ₂=0.05, the step-size factorsρ_(2,1)=2.14, β_(2,2)=1.39, β_(2,3)=1.23, β_(2,4)=1.19; the trackingperformance of the first system output is shown as RUN7 in FIG. 13 , thetracking performance of the second system output is shown as RUN7 inFIG. 14 , the first control input is shown as RUN7 in FIG. 15 , and thesecond control input is shown as RUN7 in FIG. 16 ; evaluate the controlmethod from the control performance indexes: the RMSE(e₁) of the firstsystem output, RUN7 in FIG. 13 , is 2.3072, and the RMSE(e₂) of thesecond system output, RUN7 in FIG. 14 , is 0.1683.

The eighth experiment (RUN8): the existing MIMO partial-form model-freecontrol method with the same-factor structure is adopted to control theabove two-input two-output MIMO system; set the penalty factor λ=0.05,the step-size factors ρ₁=1.5, ρ₂=ρ₃=ρ₄=1; the tracking performance ofthe first system output is RUN8 in FIG. 13 , the tracking performance ofthe second system output is RUN8 in FIG. 14 , the first control input isthe RUN8 in FIG. 15 , and the second control input is RUN8 in FIG. 16 ;evaluate the control method from the control performance indexes: theRMSE(e₁) of the first system output, RUN8 in FIG. 13 , is 2.4435, andthe RMSE(e₂) of the second system output, RUN8 in FIG. 14 , is 0.1828.

The comparison results of control performance indexes of the threeexperiments are shown in Table 2; the results of the sixth experiment(RUN6) using the inventive control method are superior to those of theseventh experiment (RUN7) using the MIMO different-factor partial-formmodel-free control method with fixed parameters, and are moresignificantly superior to those of the eighth experiment (RUN8) usingthe existing MIMO partial-form model-free control method with thesame-factor structure, which have a significant improvement, indicatingthat the inventive MIMO different-factor partial-form model-free controlmethod with parameter self-tuning has higher control accuracy, strongerstability and wider applicability.

TABLE 2 Comparison Results of The Control Performance of Coal Mill Thefirst system output The second system output RMSE(e₁) ImprovementRMSE(e₂) Improvement RUN6 2.0549 15.903% 0.1554 14.989% RUN7 2.30725.578% 0.1638 10.394% RUN8 2.4435 Baseline 0.1828 Baseline

Furthermore, the following six points should be noted in particular:

(1) In the fields of oil refining, petrochemical, chemical,pharmaceutical, food, paper, water treatment, thermal power, metallurgy,cement, rubber, machinery, and electrical industry, most of thecontrolled plants, such as reactors, distillation columns, machines,equipment, devices, production lines, workshops and factories, areessentially MIMO systems; some of these MIMO systems have the complexcharacteristics of non-minimum phase nonlinear system, which belong tothe MIMO systems that are particularly difficult to control; forexample, the continuous stirred tank reactor (CSTR), commonly used inoil refining, petrochemical, chemical, etc. is a two-input two-outputMIMO system, where the two inputs are feed flow and cooling water flow,and the two outputs are product concentration and reaction temperature;when the chemical reaction has strong exothermic effect, the continuousstirred tank reactor (CSTR) is a MIMO system with complexcharacteristics of non-minimum phase nonlinear system, which isparticularly difficult to control. In the first exemplary embodiment,the controlled plant with two inputs and two outputs also has thecomplex characteristic of non-minimum phase nonlinear system and belongsto the MIMO system that is particularly difficult to control; theinventive controller is capable of controlling the plant with highaccuracy, strong stability and wide applicability, indicating that itcan also achieve high accuracy, strong stability and wide applicabilitycontrol on complex MIMO systems such as reactors, distillation columns,machines, equipment, devices, production lines, workshops, factories,etc.

(2) In the first and second exemplary embodiments, the hardware platformfor running the inventive controller is the industrial control computer;in practical applications, according to the specific circumstance, asingle chip microcomputer controller, a microprocessor controller, afield programmable gate array controller, a digital signal processingcontroller, an embedded system controller, a programmable logiccontroller, a distributed control system, a fieldbus control system, anindustrial control system based on internet of things, or an industrialinternet control system, can also be used as the hardware platform forrunning the inventive control method.

(3) In the first and second exemplary embodiments, the j-th errore_(j)(k) is defined as the difference between the j-th desired systemoutput y_(j) ^(*)(k) and the j-th actual system output y_(j)(k), namelye_(j)(k)=y_(j) ^(*)(k)−y_(j)(k), which is only one of the methods forcalculating the j-th error; the j-th error e_(j)(k) can also be definedas the difference between the j-th desired system output y_(j) ^(*)(k+1)at time k+1 and the j-th actual system output y_(j)(k), namelye_(j)(k)=y_(j) ^(*)(k+1)−y_(j)(k); the j-th error e_(j)(k) can also bedefined by other methods whose independent variables include the j-thdesired system output and the j-th actual system output, for example,

${{e_{j}(k)} = {\frac{{y_{j}^{*}\left( {k + 1} \right)} + {y_{j}^{*}(k)}}{2} - {y_{j}(k)}}};$for the controlled plants in the first and second exemplary embodiments,all different definitions of the error function can achieve good controlperformance.

(4) The inputs of BP neural network include at least one of: theelements in set {gradient set}, and the elements in set {error set};when the inputs of BP neural network include the elements in set{gradient set}, the gradients at time k−1 are used in the firstexemplary embodiment, namely

$\left\{ {\frac{\partial{u_{1}\left( {k - 1} \right)}}{\partial\lambda_{1}},\frac{\partial{u_{1}\left( {k - 1} \right)}}{\partial\rho_{1,1}},\frac{\partial{u_{1}\left( {k - 1} \right)}}{\partial\rho_{1,2}},\frac{\partial{u_{1}\left( {k - 1} \right)}}{\partial\rho_{1,3}},\frac{\partial{u_{2}\left( {k - 1} \right)}}{\partial\lambda_{2}},\frac{\partial{u_{2}\left( {k - 1} \right)}}{\partial\rho_{2,1}},\frac{\partial{u_{2}\left( {k - 1} \right)}}{\partial\rho_{2,2}},\frac{\partial{u_{2}\left( {k - 1} \right)}}{\partial p_{2,3}}} \right\};$in practical applications, the gradients at more time can be furtheradded according to the specific situation; for example, the gradients attime k−2 can be added, namely

$\left\{ {\frac{\partial{u_{1}\left( {k - 2} \right)}}{\partial\lambda_{1}},\frac{\partial{u_{1}\left( {k - 2} \right)}}{\partial\rho_{1,1}},\frac{\partial{u_{1}\left( {k - 2} \right)}}{\partial\rho_{1,2}},\frac{\partial{u_{1}\left( {k - 2} \right)}}{\partial\rho_{1,3}},\frac{\partial{u_{2}\left( {k - 2} \right)}}{\partial\lambda_{2}},\frac{\partial{u_{2}\left( {k - 2} \right)}}{\partial\rho_{2,1}},\frac{\partial{u_{2}\left( {k - 2} \right)}}{\partial\rho_{2,2}},\frac{\partial{u_{2}\left( {k - 2} \right)}}{\partial p_{2,3}}} \right\};$when the inputs of BP neural network include the elements in set {errorset}, the error function group

$\left\{ {{e_{1}(k)},{\sum\limits_{t = 0}^{k}{e_{1}(t)}},{{e_{1}(k)} - {e_{1}\left( {k - 1} \right)}},{e_{2}(k)},{\sum\limits_{t = 0}^{k}{e_{2}(t)}},{{e_{2}(k)} - {e_{2}\left( {k - 1} \right)}}} \right\}$is used in the first and second exemplary embodiments; in practicalapplications, more error function groups can be further added to the set{error set} according to the specific situation; for example, the secondorder backward difference of the j-th error e_(jy)(k), namely{e₁(k)−2e₁(k−1)+e₁(k−2), e₂(k)−2e₂(k−1)+e₂(k−2)}, can also be added intothe inputs of BP neural network; furthermore, the inputs of BP neuralnetwork include, but is not limited to, the elements in set {gradientset} and set {error set}; for example, {u₁(k−1), u₂(k−1)} can also beadded into the inputs of BP neural network; for the controlled plants inthe first and second exemplary embodiments, the inventive controller canachieve good control performance with the increasing of the number ofinput layer nodes of BP neural network, and in most cases it canslightly improve the control performance, but at the same time itincreases the computational burden; therefore, the number of input layernodes of BP neural network can be set to a reasonable number accordingto specific conditions in practical applications.

(5) In the first and second exemplary embodiments, when updating thehidden layer weight coefficients and the output layer weightcoefficients with the objective of minimizing the system error function,all contributions of n errors are comprehensively considered in saidsystem error function

${\sum\limits_{{jy} = 1}^{n}{e_{jy}^{2}(k)}},$which is just one of the system error functions; said system errorfunction can also adopt other functions whose independent variablesinclude any one or any combination of the elements in n errors, ndesired system outputs and n actual system outputs; for example, saidsystem error function can adopt another way of

${\sum\limits_{{jy} = 1}^{n}{e_{jy}^{2}(k)}},$such as

${\sum\limits_{{jy} = 1}^{n}{\left( {{y_{jy}^{*}(k)} - {y_{jy}(k)}} \right)^{2}\mspace{14mu}{or}\mspace{14mu}{\sum\limits_{{jy} = 1}^{n}\left( {{y_{jy}^{*}\left( {k + 1} \right)} - {y_{jy}(k)}} \right)^{2}}}};$for another example, said system error function can adopt

${{\sum\limits_{{jy} = 1}^{n}{a_{jy}{e_{jy}^{2}(k)}}} + {\sum\limits_{{iu} = 1}^{m}{b_{iu}\Delta{u_{iu}^{2}(k)}}}},$where e_(jy)(k) is the jy-th error at time k,Δu_(iu)(k)=u_(iu)(k)−u_(iu)(k−1), u_(iu)(k) is the iu-th control inputat time k, a_(jy) and b_(iu) are two constants greater than or equal to0, jy and iu are two positive integers; obviously, when b_(iu) equals tozero, said system error function only considers the contribution ofe_(jy) ²(k), indicating that the objective is to minimize the systemerror and pursue high control accuracy; when b_(iu) is greater thanzero, said system error function considers the contributions of e_(jy)²(k) and Δu_(iu) ²(k) simultaneously, indicating that the objective isnot only to minimize the system error but also to minimize the varianceof control inputs, that is, to pursue high control accuracy and stablecontrol; for the controlled plants in the first and second exemplaryembodiments, all different system error functions can achieve goodcontrol performance; compared with the system error function onlyconsidering the contribution of e_(jy) ²(k), the control accuracy isslightly reduced while the handling stability is improved when thecontributions of e_(jy) ²(k) and Δu_(iu) ²(k) are taken into accountsimultaneously in the system error function.

(6) The parameters to be tuned in said MIMO different-factorpartial-form model-free control method with parameter self-tuninginclude at least one of: penalty factors λ_(i), and step-size factorsρ_(i,1), . . . , ρ_(i,L) (i=1, . . . , m); in the first exemplaryembodiment, all penalty factors λ₁, λ₂ and step-size factors ρ_(1,1),ρ_(1,2), ρ_(1,3), ρ_(2,1), ρ_(2,2), ρ_(2,3) are self-tuned in the firstexperiment to the third experiment; in the fourth experiment, only thestep-size factors ρ_(2,1), ρ_(2,2), ρ_(2,3) for the second control inputare self-tuned, while the penalty factors λ₁, λ₂ and step-size factorsρ_(1,1), ρ_(1,2), ρ_(1,3) are fixed; in practical applications, anycombination of the parameters to be tuned can be chosen according to thespecific situation; in addition, said parameters to be tuned include,but are not limited to: penalty factors λ_(i), and step-size factorsρ_(i,1), . . . , ρ_(i,L) (i=1, . . . , m); for example, said parametersto be tuned can also include the parameters for calculating theestimated value of pseudo partitioned Jacobian matrix Φ(k) for said MIMOsystem.

It should be appreciated that the foregoing is only preferredembodiments of the invention and is not for use in limiting theinvention. Any modification, equivalent substitution, and improvementwithout departing from the spirit and principle of this invention shouldbe covered in the protection scope of the invention.

The invention claimed is:
 1. A method of MIMO different-factorpartial-form model-free control with parameter self-tuning, executed ona hardware platform for controlling a controlled plant being amulti-input multi-output (MIMO) system, wherein the MIMO system having apredetermined number of control inputs and a predetermined number ofsystem outputs, said controlled plant comprises at least one of: areactor, a distillation column, a machine, a device, a set of equipment,a production line, a workshop, and a factory, said hardware platformcomprises at least one of: an industrial control computer, a single chipmicrocomputer controller, a microprocessor controller, a fieldprogrammable gate array controller, a digital signal processingcontroller, an embedded system controller, a programmable logiccontroller, a distributed control system, a fieldbus control system, anindustrial control system based on internet of things, and an industrialinternet control system, said method comprising: calculating the i-thcontrol input u₁(k) at time k as follows:${u_{i}(k)} = {{u_{i}\left( {k - 1} \right)} + \frac{\sum\limits_{j = 1}^{n}{{\phi_{j,i,1}(k)}\left( {{\rho_{i,1}{e_{j}(k)}} - {\sum\limits_{p = 2}^{L}{\rho_{i,p}\left( {\sum\limits_{{iu} = 1}^{m}{{\phi_{j,{iu},p}(k)}\Delta{u_{iu}\left( {k - p + 1} \right)}}} \right)}}} \right)}}{\lambda_{i} + {{\Phi_{1}(k)}}^{2}}}$where k is a positive integer; m is the total number of control inputsin said MIMO system, m is a positive integer greater than 1; n is thetotal number of system outputs in said MIMO system, n is a positiveinteger; i denotes the i-th of the total number of control inputs insaid MIMO system, i is a positive integer, 1≤i≤m; j denotes the j-th ofthe total number of system outputs in said MIMO system, j is a positiveinteger, 1≤j≤n; u_(i)(k) is the i-th control input at time k;Δu_(iu)(k)=u_(iu)(k)−u_(iu)(k−1), iu is a positive integer; e_(j)(k) isthe j-th error at time k, namely the j-th element in the error vectore(k)=[e₁(k), . . . , e_(n)(k)]^(T); Φ(k) is the estimated value ofpseudo partitioned Jacobian matrix for said MIMO system at time k,Φ_(p)(k) is the p-th block of Φ(k), ϕ_(j,i,p)(k) is the j-th row and thei-th column of matrix Φ_(p)(k), ∥Φ₁(k)∥ is the 2-norm of matrix Φ₁(k); pis a positive integer, 1≤p≤L; λ_(i) is the penalty factor for the i-thcontrol input; ρ_(i,p) is the p-th step-size factor for the i-th controlinput; L is the control input length constant of linearization and L isa positive integer; for said MIMO system, calculating a control inputvector u(k)=[u₁(k), . . . , u_(m)(k)]^(T) by traversing all values of iin the positive integer interval [1, m]; said method has adifferent-factor characteristic; said different-factor characteristic isthat at least one of the following L+1 inequalities holds true for anytwo unequal positive integers i and x in the positive integer interval[1, m] during controlling said MIMO system by using said method:λ_(i)≠λ_(x); ρ_(i,1)≠ρ_(x,1); . . . ; ρ_(i,L)≠ρ_(x,L) during controllingsaid MIMO system by using said method, performing parameter self-tuningon the parameters to be tuned in said mathematical formula forcalculating the control input vector u(k)=[u₁(k), . . . , u_(m)(k)]^(T)at time k; said parameters to be tuned comprise at least one of: penaltyfactors λ_(i), and step-size factors ρ_(i,1), . . . , ρ_(i,L) (i=1, . .. , m); and obtaining the system outputs from the MIMO system byadjusting the control inputs of the MIMO system based on the calculatedcontrol input vector, such that the system outputs of the MIMO systemapproach desired system outputs to be received by the hardware platform.2. The method as claimed in claim 1 wherein said parameter self-tuningadopts neural network to calculate the parameters to be tuned in themathematical formula of said control input vector u(k)=[u₁(k), . . . ,u_(m)(k)]^(T); when updating the hidden layer weight coefficients andoutput layer weight coefficients of said neural network, the gradientsat time k of said control input vector u(k)=[u₁(k), . . . ,u_(m)(k)]^(T) with respect to the parameters to be tuned in theirrespective mathematical formula are used; the gradients at time k ofu_(i)(k) (i=1, . . . , m) in said control input vector u(k)=[u₁(k), . .. , u_(m)(k)]^(T) with respect to the parameters to be tuned in themathematical formula of said u_(i)(k) comprise the partial derivativesat time k of u_(i)(k) with respect to the parameters to be tuned in themathematical formula of said u_(i)(k); the partial derivatives at time kof said u_(i)(k) with respect to the parameters to be tuned in themathematical formula of said u_(i)(k) are calculated as follows: whenthe parameters to be tuned in the mathematical formula of said u_(i)(k)include penalty factor λ_(i), the partial derivative at time k ofu_(i)(k) with respect to said penalty factor λ_(i) is:$\frac{\partial{u_{i}(k)}}{\partial\lambda_{i}} = \frac{\sum\limits_{j = 1}^{n}{{\phi_{j,i,1}(k)}\left( {{\sum\limits_{p = 2}^{L}{\rho_{i,p}\left( {\sum\limits_{{iu} = 1}^{m}{{\phi_{j,{iu},p}(k)}\Delta{u_{iu}\left( {k - p + 1} \right)}}} \right)}} - {\rho_{i,1}{e_{j}(k)}}} \right)}}{\left( {\lambda_{i} + {{\Phi_{1}(k)}}^{2}} \right)^{2}}$when the parameters to be tuned in the mathematical formula of saidu_(i)(k) include step-size factor ρ_(i,1), the partial derivative attime k of u_(i)(k) with respect to said step-size factor ρ_(i,1) is:$\frac{\partial{u_{i}(k)}}{\partial\rho_{i,1}} = \frac{\sum\limits_{j = 1}^{n}{{\phi_{\;^{j,i,1}}(k)}{e_{j}(k)}}}{\lambda_{i} + {{\Phi_{1}(k)}}^{2}}$when the parameters to be tuned in the mathematical formula of saidu_(i)(k) include step-size factor ρ_(i,p) where 2≤p≤L, the partialderivative at time k of u_(i)(k) with respect to said step-size constantρ_(i,p) is:$\frac{\partial{u_{i}(k)}}{\partial\rho_{i,p}} = {- \frac{\sum\limits_{j = 1}^{n}{{\phi_{j,i,1}(k)}\left( {\sum\limits_{{iu} = 1}^{m}{{\phi_{j,{iu},p}(k)}\Delta{u_{iu}\left( {k - p + 1} \right)}}} \right)}}{\lambda_{i} + {{\Phi_{1}(k)}}^{2}}}$putting all partial derivatives at time k calculated by said u_(i)(k)with respect to the parameters to be tuned in the mathematical formulaof said u_(i)(k) into the set {gradient of u_(i)(k)}; for said MIMOsystem, traversing all values of i in the positive integer interval [1,m] and obtaining the set {gradient of u₁(k)}, . . . , set {gradient ofu_(m)(k)}, then putting them all into the set {gradient set}; said set{gradient set} is a set comprising all sets {{gradient of u₁(k)}, . . ., {gradient of u_(m)(k) }}; said parameter self-tuning adopts neuralnetwork to calculate the parameters to be tuned in the mathematicalformula of the control input vector u(k)=[u₁(k), . . . , u_(m)(k)]^(T);the inputs of said neural network comprise at least one of: elements insaid set {gradient set}, and elements in set {error set}; said set{error set} comprises at least one of: the error vector e(k)=[e₁(k), . .. , e_(n)(k)]^(T), and error function group of element e_(j)(k) (j=1, .. . , n) in said error vector e(k); said error function group of elemente_(j)(k) comprises at least one of: the accumulation of the j-th errorat time k and all previous times ${\sum\limits_{t = 0}^{k}{e_{j}(t)}},$the first order backward difference of the j-th error e_(j)(k) at time ke_(j)(k)−e_(j)(k−1), the second order backward difference of the j-therror e_(j)(k) at time k e_(j)(k)−2e_(j)(k−1)+e_(j)(k−2), and high orderbackward difference of the j-th error e_(j)(k) at time k.
 3. The methodas claimed in claim 2 wherein said neural network is BP neural network;said BP neural network adopts a single hidden layer structure, namely athree-layer network structure, comprising an input layer, a singlehidden layer, and an output layer.
 4. The method as claimed in claim 2wherein aiming at minimizing a system error function, said neuralnetwork adopts gradient descent method to update the hidden layer weightcoefficients and the output layer weight coefficients, where thegradients are calculated by system error back propagation; independentvariables of said system error function comprise at least one of:elements in the error vector e(k)=[e₁(k), . . . , e_(n)(k)]^(T), ndesired system outputs, and n actual system outputs.
 5. The method asclaimed in claim 4 wherein said system error function is defined as${{\sum\limits_{{jy} = 1}^{n}{a_{jy}{e_{jy}^{2}(k)}}} + {\sum\limits_{{iu} = 1}^{m}{b_{iu}\Delta{u_{iu}^{2}(k)}}}},$where e_(jy)(k) is the jy-th error at time k,Δu_(iu)(k)=u_(iu)(k)−u_(iu)(k−1), u_(iu)(k) is the iu-th control inputat time k, a_(jy) and b_(iu) are two constants greater than or equal tozero, jy and iu are two positive integers.
 6. The method as claimed inclaim 1 wherein said j-th error e_(j)(k) at time k is calculated by thej-th error function; independent variables of said j-th error functioncomprise the j-th desired system output and the j-th actual systemoutput.
 7. The method as claimed in claim 6 wherein said j-th errorfunction adopts at least one of: e_(j)(k)=y_(j) ^(*)(k)−y_(j)(k),e_(j)(k)=y_(j) ^(*)(k+1)−y_(j)(k), e_(j)(k)=y_(j)(k)−y_(j) ^(*)(k), ande_(j)(k)=y_(j)(k)−y_(j) ^(*)(k+1), where y_(j) ^(*)(k) is the j-thdesired system output at time k, y_(j) ^(*)(k+1) is the j-th desiredsystem output at time k+1, and y_(j)(k) is the j-th actual system outputat time k.